最快检测出最少的无序时间
Quickest detection of a minimum of disorder times
摘要：
研究了多源最快检测问题。假设存在两个独立的泊松过程$X^{1}$和$X^{2}$，其无序时间分别为$θ_{1}$和$θ_{2}$;即，$X^1$和$X^2$的强度分别在随机的不可观测时间$θ_1$和$θ_2$发生变化。$θ_1$和$θ_2$相互独立，呈指数分布。定义$θ\triangleq θ_1 \wedge θ_2=\min\{θ_{1},θ_{2}\}$。对于任何相对于由观测生成的过滤可测量的停止时间$τ$，定义形式为\[ R_τ=\mathbb{P}(τ<θ)+c \mathbb{E}[(τ-θ)^{+}], \]的罚函数，其中$c>0$和$(τ-θ)^{+}$是$τ-θ$的正部分。感兴趣的是找到使上述性能指标最小化的停止时间$τ$。由于观测值$X^{1}$和$X^{2}$都揭示了关于无序时间$θ$的信息，因此即使是这个简单的问题也比分别解决$X^{1}$和$X^{2}$的无序问题更复杂。这个问题是制定在三维充分统计方面，相应的最佳停止问题进行检查。还解决了一个二维最优停时问题，其最优停时在一定参数范围内与原问题的最优停时一致。这个问题的价值函数作为原问题的价值函数的紧上界。这两个解决方案的特点是迭代合适的功能操作。
英文摘要：
A multi-source quickest detection problem is considered. Assume there are two independent Poisson processes $X^{1}$ and $X^{2}$ with disorder times $θ_{1}$ and $θ_{2}$, respectively; that is, the intensities of $X^1$ and $X^2$ change at random unobservable times $θ_1$ and $θ_2$, respectively. $θ_1$ and $θ_2$ are independent of each other and are exponentially distributed. Define $θ\triangleq θ_1 \wedge θ_2=\min\{θ_{1},θ_{2}\}$ . For any stopping time $τ$ that is measurable with respect to the filtration generated by the observations define a penalty function of the form \[ R_τ=\mathbb{P}(τ<θ)+c \mathbb{E}[(τ-θ)^{+}], \] where $c>0$ and $(τ-θ)^{+}$ is the positive part of $τ-θ$. It is of interest to find a stopping time $τ$ that minimizes the above performance index. Since both observations $X^{1}$ and $X^{2}$ reveal information about the disorder time $θ$, even this simple problem is more involved than solving the disorder problems for $X^{1}$ and $X^{2}$ separately. This problem is formulated in terms of a three dimensional sufficient statistic, and the corresponding optimal stopping problem is examined. A two dimensional optimal stopping problem whose optimal stopping time turns out to coincide with the optimal stopping time of the original problem for some range of parameters is also solved. The value function of this problem serves as a tight upper bound for the original problem's value function. The two solutions are characterized by iterating suitable functional operators.
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